Witt theorems for Grassmannians and Lie Incidence Geometry.
Bruce Cooperstein, University of California, Santa Cruz.
The concept of singular independence and local independence will be defined for subgraphs of the collinearity graph of a Lie incidence geometry among which are the Grassmannians. I will also introduce the notion of a parabolic subgeometry of a Lie incidence geometry and a parabolic subgraph of the shadow of an apartment. For a Grassmannian G(n,k) a parabolic subgeometry is nothing more than the subgeometry of k-spaces incident with a flag of the underlying PG(n-1).
It will be shown that a subgraph of a Grassmannian geometry or a half spin geometry which is singular independent and isomorphic to a parabolic subgraph is parabolic. It will also be proved that a subgraph of a spin geometry which is locally independent and isomorphic to a parabolic subgraph is parabolic. As a corollary it will be proved that any subgeometry which is isomorphic to a parabolic subgeometry is, in fact, parabolic and all such subspaces will be classified.